Generalization Error Bounds for Picard-Type Operator Learning in Nonlinear Parabolic PDEs

2026-05-11

AI Summary

This work addresses the learning of solution operators for nonlinear parabolic partial differential equations under finite resolution constraints by proposing an abstract state-transition operator learning framework based on Duhamel–Picard iteration. The method decouples Picard iteration depth from generalization error—a first in the literature—and theoretically demonstrates that increasing iteration depth effectively reduces truncation error without causing estimation error to explode, thereby enabling stable long-horizon rollouts. An entropy-based analysis yields implementation-agnostic generalization error bounds, and the framework is instantiated using Fourier neural operators. Experiments on nonlinear heat equations over the torus show that the proposed Picard-type operator achieves high accuracy and robustness in both multiscale and long-time predictions.

Abstract

Operator learning for partial differential equations (PDEs) aims to learn solution operators on infinite-dimensional function spaces from finite-resolution data. In this setting, it is important for the learned model to be discretization-invariant, or resolution-robust, and to reflect PDE-specific structure. It is therefore natural to ask how such structure should be encoded in the model architecture, hypothesis class, or learning procedure. In this paper, we study operator learning for solution operators of nonlinear parabolic PDEs based on Duhamel--Picard iteration. We formulate Picard iteration as an abstract state-transition model and present a theoretical framework for Picard-type operator learning. We derive implementation-agnostic generalization error bounds that separate the implementation error from the estimation error associated with the abstract state-transition model induced by Picard iteration. A key consequence is that increasing the Picard depth reduces the Picard truncation error without causing an unbounded growth of the entropy-based estimation error. We also extend the analysis to long-time prediction by rolling out the same learned local model over successive time blocks. Finally, we illustrate the theory for nonlinear heat equations on the torus using a Picard-type Fourier neural operator as a concrete implementation.

Problem

• operator learning
• nonlinear parabolic PDEs
• generalization error
• discretization-invariance
• Picard iteration

Innovation

• Picard iteration
• operator learning
• generalization error bounds
• nonlinear parabolic PDEs
• resolution-robustness